Equation 1.2 where k = Boltzmann’s constant T = temperature in degrees Kelvin r = radius of the molecule η= viscosity of the medium
According to eqs. 1.1 and 1.2 , the rate of diffusion will be faster for small molecules than for large molecules. In addition, diffusion rates are high at elevated temperatures, in the presence of large concentration gradients, and when diffusion occurs in a low-viscosity medium. With all other variables held constant, the rate of diffusion is linearly related to the concentration gradient.
Fick’s equation can also be applied to the diffusion of molecules across a barrier, such as a lipid bilayer. When applied to the diffusion of a molecule across a bilayer, the diffusion coefficient (D) incorporates the properties of the bilayer and especially the ability of the molecule to diffuse through the bilayer. To quantify the interaction of the molecule with the bilayer, the term partition coefficient (β) is used. For a molecule that “dissolves” equally in the fluid bathing the lipid bilayer (e.g., water) and in the lipid bilayer, β= 1. If the molecule dissolves more easily in the lipid bilayer, β> 1; and if it dissolves less easily in the lipid bilayer, β< 1. For a simple lipid bilayer, the more lipid soluble the molecule is, the larger the partition coefficient is, and thus the diffusion coefficient—therefore the rate of diffusion of the molecule across the bilayer—is greater. In this situation, ΔC represents the concentration difference across the membrane, A is the membrane area, and ΔX is the thickness of the membrane.
Another useful equation for quantitating the diffusion of molecules across the plasma membrane (or any membrane) is as follows:
Equation 1.3
J =−PC( i − Co), where J = the flux or rate of diffusion across the membrane P = is the permeability coefficient Ci = concentration of the molecule inside the cell Co = the concentration of the molecule outside the cell
This equation is derived from Fick’s equation ( eq. 1.1 ). P incorporates D, ΔX, A, and the partition coefficient (β). P is expressed in units of velocity (e.g., centimeters per second), and C the units of moles/cm3. Thus the units of flux are moles per square centimeter per second (mol/cm2/ sec). Values for P can be obtained experimentally for any molecule and bilayer.
As noted, the phospholipid portion of the plasma membrane represents an effective barrier to many biologically important molecules. Consequently, diffusion through the lipid phase of the plasma membrane is not an efficient process for movement of these molecules across the membrane. It has been estimated that for a cell 20 µm in diameter, with a plasma membrane composed only of phospholipids, dissipation of a urea gradient imposed across the membrane would take approximately 8 minutes. Similar gradients for glucose and amino acids would take approximately 14 hours to dissipate, whereas ion gradients would take years to dissipate.
As noted previously, the vast majority of biologically important molecules cross cell membranes via specific membrane transporters, rather than by diffusing through the lipid portion of the membrane. Nevertheless, eq. 1.3 can be and has been used to quantitate the diffusion of molecules across many biological membranes. When this is done, the value of the permeability coefficient (P) reflects the properties of the pathway (e.g., membrane transporter or, in some cases, multiple transporters) that the molecule uses to cross the membrane.